In U.S. Pat. No. 5,091,320, Aspnes et al. described a method for real-time control of the growth of thin films of compound semiconductors by use of ellipsometry. This patent is incorporated herein by reference, and the same work was reported by Aspnes et al. in "Optical control of growth of Al.sub.x Ga.sub.1-x As by organometallic molecular beam epitaxy," Applied Physics Letters, volume 57, 1990, pp.2707-2709 and in "Growth of Al.sub.x Ga.sub.1-x As parabolic quantum wells by real-time feedback control of composition," ibid., volume 60, pp. 1244-1246. Quinn has disclosed a method of aligning the ellipsometer in U.S. patent application, Ser. No. 07/723,580, filed Jul. 1, 1991. The general operation of the ellipsometric feedback control of thin-film growth is illustrated in FIG. 1. A thin film 10 of AlGaAs is grown on a GaAs substrate 12 by organo-metallic molecular beam epitaxy (OMMBE) performed within a growth chamber 14 pumped to low pressures by a pump 16. Arsine (AsH.sub.3) is supplied to a cracking unit 18 where it is cracked into molecular arsenic (As.sub.2) which irradiates the heated substrate 12 on which the thin film 10 is growing. Gas-entrained triethylgallium and triethylaluminum or triisobutylaluminum are supplied to ports 20 and 22. The respective gases irradiate the hot thin film 10, upon the hot surface of which the triethylgallium and the triethylaluminum crack into gallium and aluminum. The arsenic, gallium, and aluminum then chemically combine to epitaxially form the thin film 10 with the crystalline orientation of the substrate. The amounts of the three constituents are controlled by respective valves 24, 26, and 28. The chamber 14 is maintained with an overpressure of arsenic so that the alloying fraction x for Al.sub.x Ga.sub.1-x As is determined by the relative amounts of triethylgallium and triethylaluminum.
An ellipsometer continuously monitors the thin film 10 while it is being grown. In the ellipsometer, an incident beam of light 30 from a wide-band light source 32 has its linear polarization angle continually changed by a rotating Rochon prism 34. A beam 36 reflected from the thin film 10 is focused in a monochromator 38 through a fixed analyzer prism, and the intensity of its monochromatic output is detected by a photomultiplier tube 40. A computer system 42 receives the intensity of the monochromatic light for multiple sampling periods during at least one complete rotation of the polarizer 34 and uses this data to calculate the ellipsometric parameters .psi. and .DELTA.. It thus establishes the complex reflectance ratio EQU .rho.=tan .psi.e.sup.i.DELTA. =r.sub.p /r.sub.s, (1)
where r.sub.p and r.sub.s are the complex reflectances for p- and s-polarized light, respectively. The complex reflectance coefficients are ratios of complex field coefficients for different polarizations and are not themselves measured.
Aspnes et al. rely on the fact that the ellipsometrically measured parameters can be related through the complex reflectance ratio .rho. to the composition of the thin film 10 to compare the ellipsometrically determined composition to the target composition and to accordingly adjust the Al valve 28 in real time so as to correct the composition being deposited. Their formalism relies on the fact that the complex reflectances themselves, and therefore their ratio, can be related within some simple models to material properties, such as material composition or the dielectric constant .epsilon., which is directly related to the composition. That is, they use ellipsometry to continuously physically characterize the deposited film and to readjust in real time the growth conditions to thereby achieve a film of the desired characteristic. In particular, they assume a three-phase model consisting of a homogeneous substrate having a dielectric constant .epsilon..sub.s, a homogeneous thin film having a dielectric constant .epsilon..sub.o, and a homogeneous ambient having a dielectric constant .epsilon..sub.a, which in the case of air or vacuum is equal to one. Closed-form solutions exist within the three-phase model for the complex reflectances r.sub. p and r.sub.s in terms of the complex dielectric function .epsilon..sub.o of the film and its thickness t. They then expand this closed-form to first order in t and observe that, as the film thickness t increases, both complex reflectances follow a spiral EQU Z=e.sup.i2k.sbsp.o.sup.t ( 2)
in the complex plane. The locus spirals inwardly from the complex reflectance r.sub.sa for the bare substrate to the complex reflectance r.sub.oa for an optically thick film. In this equation, ##EQU1## where .phi. is the angle from the normal for both the incident and reflected beams, .omega. is the frequency of the light, and c is the speed of light. Specifically, closed-form equations exist in this model for both r.sub.p and for r.sub.s, both of the form EQU r=r.sub.oa +[r.sub.sa -r.sub.oa ]e.sup.i2k.sbsp.o.sup.t. (4)
In these equations, the material information of the homogeneous film is contained in k.sub.o and r.sub.oa. Because ellipsometric measurements do not permit r.sub.s and r.sub.p to be individually determined, Aspnes et al. work with the complex reflectance ratio which is approximated by EQU .rho..apprxeq..rho..sub.oa +[.rho..sub.sa -.rho..sub.oa ]e.sup.i2k.sbsp.o.sup.t. (5)
However, the form of Equation (5) is accurate only to the extent that EQU .vertline.(r.sub.sa.sup.2 -r.sub.oa.sup.2)e.sup.i2k.sbsp.o.sup.t .vertline.&lt;&lt;.vertline.r.sub.oa.sup.2 .vertline.. (6)
In fact, they work within the formalism of the pseudo-dielectric function &lt;.epsilon.&gt; which is the complex dielectric constant seen by an ellipsometer. That is, it assumes a two-phase model of a homogeneous sample for which material parameters can be analytically related to ellipsometric data, for example, ##EQU2## where &lt;.epsilon.&gt; would be the uniform dielectric constant of the homogeneous sample. They then assume that the complex pseudo-dielectric function &lt;.epsilon.(t)&gt; will follow the same complex spiral as a function of the thickness of a uniform film EQU &lt;.epsilon.(t)&gt;=.epsilon..sub.o +(.epsilon..sub.s -.epsilon..sub.o)e.sup.i2k.sbsp.o.sup.t. (8)
In essence, this is equivalent to performing another first-order expansion, this one performed on Equation (7). Therefore, as the film thickness t increases from zero to a large value, an ellipsometrically measured complex dielectric function will follow Equation (8) assuming the spiral dependence on t. The material information is now contained in k.sub.o and .epsilon..sub.o. Aspnes et al. then expand the spiral dependence to first order in a growth increment .DELTA.t to obtain the complex dielectric constant of that growth increment ##EQU3## which is the dielectric function of the film. In this equation, &lt;.epsilon.(t)&gt; is the ellipsometrically measured pseudo-dielectric function at some thickness t and .DELTA.&lt;.epsilon.(t)&gt;/.DELTA.t is its differential (derivative) over .DELTA.t. Because of the dependence of k.sub.o on .epsilon..sub.o, Equation (9) is a cubic equation in .epsilon..sub.o which can be solved for the value of .epsilon..sub.o. However, in the earlier reported work, the computer system 42 used a linear approximation of Equation (9) for .epsilon..sub.o. The computer system 42 then compares the measured dielectric constant of the thin film to a target value representing the desired composition and accordingly changes the Al valve 28. Further details of averaging periods, time constants, and other calculational procedures can be found in the patent.
Although the technique described above provides for vastly improved compositional control, it suffers some disadvantages. Its convergence to a final compositional value is felt to be too slow. Its convergence is controlled within a model assuming a homogeneous film, which is of course incorrect insofar as any correction is required. More generally, situations exist where the three-phase model is totally inappropriate. For structures having a very thin buried layer, that is, one that is not optically thick, the three-phase model is incorrect for a later grown layer. Furthermore, it may be desirable to compositionally grade an interface or an entire layer, for example, a quantum-well structure having a parabolically shaped potential well.
Theoretically, equations similar to those above could be numerically integrated with experimentally determined data as the structure is being grown. In fact, this is the standard approach for analyzing optical data taken on depositing thin films. However, false-data calculations show that when such analyses are done for thickness increments as small as those required for real time control, round-off errors, to say nothing about experimental noise, create mathematical instabilities that doom this approach.